Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras

Let X be a Gorenstein normal 3-fold satisfying (ELF) with local rings which are at worst isolated hypersurface (e.g. terminal) singularities. By using the singular derived category D_{sg}(X) and its idempotent completion, we give necessary and sufficient categorical conditions for X to be Q-factorial and complete locally Q-factorial respectively. We then relate this information to maximal modification algebras(=MMAs), introduced in [IW10], by showing that if an algebra A is derived equivalent to X as above, then X is Q-factorial if and only if A is an MMA. Thus all rings derived equivalent to Q-factorial terminalizations in dimension three are MMAs. As an application, we extend some of the algebraic results in Burban-Iyama-Keller-Reiten [BIKR] and Dao-Huneke [DH] using geometric arguments.Comment: Very minor changes, 24 page

Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities

In this paper we define and study triangulated categories in which the Homspaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If C is such a category, we say that C is Calabi–Yau with dim C ≤ 1. We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories CM R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of CM R and both partial crepant resolutions and Q-factorial terminalizations of Spec R, and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local cAn singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of CM R, we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [7, 10], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k D C we obtain many autoequivalences of the derived category of the Q-factorial terminalizations of Spec R

Contractions and deformations

Suppose that f is a projective birational morphism with at most one-dimensional fibres between d-dimensional varieties X and Y, satisfying ${\bf R}f_* \mathcal{O}_X = \mathcal{O}_Y$. Consider the locus L in Y over which f is not an isomorphism. Taking the scheme-theoretic fibre C over any closed point of L, we construct algebras $A_{fib}$ and $A_{con}$ which prorepresent the functors of commutative deformations of C, and noncommutative deformations of the reduced fibre, respectively. Our main theorem is that the algebras $A_{con}$ recover L, and in general the commutative deformations of neither C nor the reduced fibre can do this. As the d=3 special case, this proves the following contraction theorem: in a neighbourhood of the point, the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable.Comment: Minor changes following referee comments. 22 page

Flops and Clusters in the Homological Minimal Model Program

Suppose that $f\colon X\to\mathrm{Spec}\, R$ is a minimal model of a complete local Gorenstein 3-fold, where the fibres of $f$ are at most one dimensional, so by [VdB1d] there is a noncommutative ring $\Lambda$ derived equivalent to $X$. For any collection of curves above the origin, we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of $\Lambda$ is finite dimensional, improving a result of [DW2]. We further show that the mutation functor of [S6][IW4] is functorially isomorphic to the inverse of the Bridgeland--Chen flop functor in the case when the factor of $\Lambda$ is finite dimensional. These results then allow us to jump between all the minimal models of $\mathrm{Spec}\, R$ in an algorithmic way, without having to compute the geometry at each stage. We call this process the Homological MMP. This has several applications in GIT approaches to derived categories, and also to birational geometry. First, using mutation we are able to compute the full GIT chamber structure by passing to surfaces. We say precisely which chambers give the distinct minimal models, and also say which walls give flops and which do not, enabling us to prove the Craw--Ishii conjecture in this setting. Second, we are able to precisely count the number of minimal models, and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity. Third, we prove a bijective correspondence between maximal modifying $R$-module generators and minimal models, and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry. This lifts the Auslander--McKay correspondence to dimension three.Comment: 58 pages. Last update had an old version of Section 4, no other changes. Final version, to appear Invent. Mat